Analysis/Advanced Calculus ?

Question

Suppose f : D → R with x0 an accumulation point of D. Assume L1 and L2 are limits of f at x0. Prove L1 = L2

(definition of the limit of a function)

Answer 1

2 figure out on rest own

Answer 2

This is almost identical to the other question about one function always being smaller and showing the limit of that is smaller than the limit of the other function.
I'm not going to go through all the details this time, just a general outline:

Assume the limits are not equal. Choose ε to be half the gap between them. Then writing out the definition of each limit will show you that for x close to x0, f(x) will be within ε of both L1 and L2, but thats impossible, since the distance between L1 and L2 is at least 2*ε.